Levenshtein Codes Research Articles

DNA-QLC: an efficient and reliable image encoding scheme for DNA storage

BackgroundDNA storage has the advantages of large capacity, long-term stability, and low power consumption relative to other storage mediums, making it a promising new storage medium for multimedia information such as images. However, DNA storage has a low coding density and weak error correction ability.ResultsTo achieve more efficient DNA storage image reconstruction, we propose DNA-QLC (QRes-VAE and Levenshtein code (LC)), which uses the quantized ResNet VAE (QRes-VAE) model and LC for image compression and DNA sequence error correction, thus improving both the coding density and error correction ability. Experimental results show that the DNA-QLC encoding method can not only obtain DNA sequences that meet the combinatorial constraints, but also have a net information density that is 2.4 times higher than DNA Fountain. Furthermore, at a higher error rate (2%), DNA-QLC achieved image reconstruction with an SSIM value of 0.917.ConclusionsThe results indicate that the DNA-QLC encoding scheme guarantees the efficiency and reliability of the DNA storage system and improves the application potential of DNA storage for multimedia information such as images.

On Prefixed Varshamov-Tenengolts Codes for Segmented Edit Channels

The prefixed Varshamov-Tenengolts (VT) codes, which are subsets of VT codes with predetermined prefixes, can be used for error correction over segmented edit channels. In this paper, we investigate the construction and analysis of this class of codes. First, we derive upper bounds on the size of zero-error codes for segmented edit channels with segment-by-segment decoding. Second, we establish a one-to-one correspondence between prefixed VT codes and Levenshtein codes. Based on this relation, we can obtain explicit formulas on the size of prefixed VT codes via the existing results on the size of Levenshtein codes. Third, we construct a new zero-error prefixed VT code and show that the size of the constructed code is strictly larger than that of the existing prefixed VT code for the segmented deletion channel. Finally, an efficient systematic encoding method of prefixed VT codes is proposed for the segmented edit channels.

Deletion correcting codes meet the Littlewood–Offord problem

In this paper, we make a novel connection between information theory and additive combinatorics; more specifically, between deletion/insertion correcting codes and the celebrated Littlewood–Offord problem. We see how results from one area can have an impact on the other area and vice versa. In particular, a result on the Littlewood–Offord problem gives a nice upper bound for the size of the Levenshtein code and the Helberg code (and possibly other variants of these codes). Also, a recent result on the deletion correcting codes gives a modular analogue of the Littlewood–Offord problem which generalizes the results of Vaughan and Wooley (Q J Math Oxf Ser (2) 42(2):379–386, 1991) (obtained using tools from analytic number theory and properties of exponential sums) and of Griggs (Bull Am Math Soc (N.S.) 28:329–333, 1993) (obtained using a combinatorial argument). This novel connection might opens up new doors to research in these or other related areas.

Explicit Formulas for the Weight Enumerators of Some Classes of Deletion Correcting Codes

We introduce a general class of codes which includes several well-known classes of deletion/insertion correcting codes as special cases. For example, the Helberg code, the Levenshtein code, the Varshamov--Tenengolts code, and most variants of these codes including most of those which have been recently used in studying DNA-based data storage systems are all special cases of our code. Then, using a number theoretic method, we give an explicit formula for the weight enumerator of our code which in turn gives explicit formulas for the weight enumerators and so the sizes of all the aforementioned codes. We also obtain the size of the Shifted Varshamov--Tenengolts code. Another application which automatically follows from our result is an explicit formula for the number of binary solutions of an arbitrary linear congruence which, to the best of our knowledge, is the first result of its kind in the literature and might be also of independent interest. Our general result might have more applications/implications in information theory, computer science, and mathematics.

Indel-correcting DNA barcodes for high-throughput sequencing

Many large-scale, high-throughput experiments use DNA barcodes, short DNA sequences prepended to DNA libraries, for identification of individuals in pooled biomolecule populations. However, DNA synthesis and sequencing errors confound the correct interpretation of observed barcodes and can lead to significant data loss or spurious results. Widely used error-correcting codes borrowed from computer science (e.g., Hamming, Levenshtein codes) do not properly account for insertions and deletions (indels) in DNA barcodes, even though deletions are the most common type of synthesis error. Here, we present and experimentally validate filled/truncated right end edit (FREE) barcodes, which correct substitution, insertion, and deletion errors, even when these errors alter the barcode length. FREE barcodes are designed with experimental considerations in mind, including balanced guanine-cytosine (GC) content, minimal homopolymer runs, and reduced internal hairpin propensity. We generate and include lists of barcodes with different lengths and error correction levels that may be useful in diverse high-throughput applications, including >106 single-error-correcting 16-mers that strike a balance between decoding accuracy, barcode length, and library size. Moreover, concatenating two or more FREE codes into a single barcode increases the available barcode space combinatorially, generating lists with >1015 error-correcting barcodes. The included software for creating barcode libraries and decoding sequenced barcodes is efficient and designed to be user-friendly for the general biology community.

Efficient Low-Redundancy Codes for Correcting Multiple Deletions

We consider the problem of constructing binary codes to recover from $k$ –bit deletions with efficient encoding/decoding, for a fixed $k$ . The single deletion case is well understood, with the Varshamov–Tenengolts–Levenshtein code from 1965 giving an asymptotically optimal construction with $\approx ~2^{n}/n$ codewords of length $n$ , i.e., at most $\log n$ bits of redundancy. However, even for the case of two deletions, there was no known explicit construction with redundancy less than $n^{\Omega (1)}$ . For any fixed $k$ , we construct a binary code with $c_{k} \log n$ redundancy that can be decoded from $k$ deletions in $O_{k}(n \log ^{4} n)$ time. The coefficient $c_{k}$ can be taken to be $O(k^{2} \log k)$ , which is only quadratically worse than the optimal, non-constructive bound of $O(k)$ . We also indicate how to modify this code to allow for a combination of up to $k$ insertions and deletions. We also note that among linear codes capable of correcting $k$ deletions, the $(k+1)$ -fold repetition code is essentially the best possible.

Insertion/Deletion Detecting Codes and the Boundary Problem

Insertion/deletion detecting codes were introduced by Konstantinidis In this paper we define insertion/deletion detecting codes in a slightly different manner, and based on this definition, we introduce multiple deletion and multiple insertion detecting codes. It is shown that these codes, which are systematic, are optimal in the sense that there exists no other systematic multiple deletion (insertion) detecting codes with a better rate. One of the limitations of number-theoretic code constructions intended to correct insertion/deletion errors, e.g., the Levenshtein code, is that they require received codeword boundaries to be known in order to successfully decode. In literature, a number of schemes have been proposed to deal with this problem. We show how insertion/deletion detecting codes as presented in this paper can be used to improve and/or extend some of these schemes.

On Helberg's Generalization of the Levenshtein Code for Multiple Deletion/Insertion Error Correction

A proof that the Helberg code is capable of correcting multiple deletion/insertion errors is presented. This code is a generalization of the number-theoretic Levenshtein code which is capable of correcting a single deletion/insertion. However, apart from exhaustive testing of short codes, no proof was hitherto given to verify that the Helberg code is indeed capable of correcting multiple deletions and insertions.

Binary permutation sequences as subsets of Levenshtein codes, spectral null codes, run-length limited codes and constant weight codes

We investigate binary sequences which can be obtained by concatenating the columns of (0,1)-matrices derived from permutation sequences. We then prove that these binary sequences are subsets of a surprisingly diverse ensemble of codes, namely the Levenshtein codes, capable of correcting insertion/deletion errors; spectral null codes, with spectral nulls at certain frequencies; as well as being subsets of run-length limited codes, Nyquist null codes and constant weight codes.

Data Coding by Linear Forms of Numerical Sequences

This paper proposes to code integers and binary trees by linear forms of (au_{n-1}+bu_{n})-type, where u_{n-1} and u_{n-1} are adjacent members of some integer sequence. New prefix codes of integers are obtained. Such codes have better characteristics than the well-known Levenshtein code. Data compression by means of coding by linear forms is considered. Linear Fibonacci forms are studied in detail.

A new recursive universal code of the positive integers

A new recursive universal code of the positive integers is proposed, in which any given sequence can be used as a delimiter of codeword while bit 0 is used as a delimiter in known universal codes, e.g., Levenshtein code, Elias /spl omega/ code, Even-Rodeh code, Stout code, Bentley-Yao code, etc. The codeword length of the proposed code is shorter than log/sub 2//sup n/ n in almost all of sufficiently large positive integers although the known codes are longer than log/sub 2//sup n/ n for any positive integer n.

On the rectangularity of nonlinear block codes

We give simple sufficient conditions for a code to be rectangular and show that large families of well-known nonlinear codes are rectangular. These include Hadamard (1893), Levenshtein (1964), Delsarte-Goethals (1975), Kerdock (1972), and Nordstrom-Robinson (1967) codes. Being rectangular, each of these codes has a unique minimal trellis that can be used for soft-decision maximum-likelihood decoding.

Insertion/deletion correction with spectral nulls

Levenshtein (1966) proposed a class of single insertion/deletion correcting codes, based on the number-theoretic construction due to Varshamov and Tenengolt's (1965). We present several interesting results on the binary structure of these codes, and their relation to constrained codes with nulls in the power spectral density function. One surprising result is that the higher order spectral null codes of Immink and Beenker (1987) are sub-codes of balanced Levenshtein codes. Other spectral null sub-codes with similar coding rates, may also be constructed. We furthermore present some coding schemes and spectral shaping markers which alleviate the fundamental restriction on Levenshtein's codes that the boundaries of each codeword should be known before insertion/deletion correction can be effected.